Summary: We consider a class of (constrained) minimization problems: find \(x_ 0\in A\) such that \(f(x_ 0)=\min\) \(\{\) f(x): \(x\in A\}\). Here A is an arbitrary non-empty closed subset of a complete metric space X and \(f: X\to R\) is an arbitrary continuous real-valued function bounded from below. Obviously, every problem of this class is determined by the pair (A,f). The set of all these pairs is endowed with a natural complete metric.
Our aim is to show that the ’majority’ of the problems from the above class are well-posed (i.e. there is uniqueness and continuous dependence of solutions on A and f; the well-posedness in this article means that both Hadamard and Tykhonov well-posedness are fulfilled). Here the ’majority’ is understood in the Baire category sense, viz. the set of well-posed problems contains a dense \(G_{\delta}\)-subset of all the complete metric space, i.e. its complement is of first Baire category and is considered to be a small set. Several classes of convex optimization problems are investigated to be a small set. Several classes of convex optimization problems are investigated in the same direction.